A right triangle is a triangle that has one angle measuring 90 degrees. In trigonometry, right triangle problems involve using the Pythagorean theorem or trigonometric functions (sine, cosine, and tangent) to solve for missing sides or angles of a right triangle.

Key Points:

• A right triangle has one 90-degree angle.
• The side opposite the 90-degree angle is the hypotenuse, the longest side of the triangle.
• The other two sides are called the legs (one is the opposite side, and the other is the adjacent side to the angle of interest).

The Pythagorean Theorem:

The Pythagorean theorem is used to find the length of a side in a right triangle when the lengths of the other two sides are known. It states: a² + b² = c² where:

• a and b are the lengths of the legs of the triangle.
• c is the length of the hypotenuse (the side opposite the right angle).

Example 1: Finding the Hypotenuse

Consider a right triangle where the lengths of the legs are 3 units and 4 units. To find the hypotenuse, we use the Pythagorean theorem:

• 3² + 4² = c²
• 9 + 16 = c²
• 25 = c²
• c = √25 = 5

Therefore, the length of the hypotenuse is 5 units.

Example 2: Using Trigonometric Functions to Find a Missing Side

Consider a right triangle with an angle of 30 degrees, and the length of the hypotenuse is 10 units. To find the length of the side opposite the angle (using sine), we use the formula:

• sin(30°) = opposite / hypotenuse
• 0.5 = opposite / 10
• opposite = 0.5 × 10 = 5

Therefore, the length of the side opposite the 30° angle is 5 units.

Example 3: Using Trigonometric Functions to Find a Missing Side

Consider a right triangle with an angle of 45 degrees, and the length of the adjacent side is 7 units. To find the length of the opposite side (using tangent), we use the formula:

• tan(45°) = opposite / adjacent
• 1 = opposite / 7
• opposite = 1 × 7 = 7

Therefore, the length of the side opposite the 45° angle is 7 units.

Summary:

Right triangle problems involve using the Pythagorean theorem or trigonometric functions (sine, cosine, and tangent) to solve for unknown sides or angles. These tools are crucial for understanding and solving real-world problems in geometry and physics that involve right triangles.

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